I don't get why there are two different programs in a minimal install to install software. Don't they do the same thing? Is there a big difference? I have read everywhere to use aptitude over apt-get but I still don't know the difference

The official tool that is currently used within the Debian installer and recommended in the release notes is aptitude.

Aptitude offers a curses interface (when run without any parameter) and a command line interface that can do almost everything that apt-cache/apt-get does. It also has a better dependency resolver that let you browse between multiple solutions. Even when using the command line version, you can interact with the proposed solution and give supplementary orders or hints (like installing or removing a package that is recommended by another one).

But aptitude is based on the libapt library (it's not a direct wrapper of dpkg) and as such it depends on the apt package so you can't have aptitude installed without apt-get (which is also in the apt package).

aptitude remebers which packages you deliberately installed and which have been automatically installed to satisfy dependencies. Upon removal of a package aptitude will remove those automatically installed packages to keep your system tidy. apt-get isn't that smart.

aptitude will quickly become an indispensable tool once you use different repositories with different versions of a package. Imagine you are working with backports.org or inofficial repositories like debian-multimedia.org. Then aptitude will show (at the bottom of the details page of a certain package) the available version number and allow you to select one.

If you get into a situation where programs conflict due to such usage of inofficial repositories you will see a "Broken: ..." display on the top. You can type l~b (limit / flag / broken) and just get a list of packages that conflict with each other. Makes it much easier to resolve the conflict manually.

Futhermore you can use "l" to limit the package list. Imagine you are looking for packages that start with "openoffice" then press "l" and type "^openoffice" (it's a regular expression) and you will see just those packages. Choose a "flat package list" from the menu and you will get reach your goal quickly.

The complete handbook is very large offering you many more filters and possibilities. Even if you don't know them aptitude is a handy replacement for apt-get and you can use it with common parameters like

Actually, as of Lenny, apt-get also tracks packages installed only as dependencies of something you explicitly asked for. I use aptitude myself, but apt-get autoremove now removes packages that were installed as dependencies of a package that has since been removed.
–
TelemachusJun 7 '09 at 20:41

as you know when you install a package if it needs other packages (dependencies) they will be downloaded and installed too. aptitude tracks, at the moment of install, the packages that come with the installed one so when you remove the latter aptitude will ask you for the removal of the other (now unused) "orphaned" dependencies. apt-get will keep them in your system silently if you remove the "main" package.

Aside from providing a pretty console UI when you run aptitude with no arguments, it combines the various apt-* commands (and dselect) into one utility..

To search for a package and install it, using apt-get:

apt-cache search somepkg
apt-get install somepkg

..but with aptitude it's the same command:

aptitude search somepkg
aptitude install somepkg

aptitude does have some additional features, like aptitude changelog somepkg and things like holding packages (to stop them being upgraded) - nothing you couldn't achieve via other commands/methods, it's just more unified and nice to use.

The biggest difference is that aptitude is designed to be interactive where the apt tools are better suited for scripting work. You can use the latter, i.e. the apt tools, interactively just fine with a lot more effort to get the same results that aptitude provides. If you had to settle on just one tool or the other for some reason then you need to decide how much effort you want to put into your package management.

One of the serious advantages of aptitude is its ability to calculate dependencies. Firstly it can handle complicated inter-dependencies without intervention by yourself, which often apt-get fails on. But if you don't like the solution it has presented you (and it will score its own solution to demonstrate how good it thinks it is), you can ask it to go away and come up with another solution to resolve the conflict.

It gives you a Text based user interface, you can use keybord to search/navigate etc. see information on dependencies, reverse dependencies, all available versions of a package, and you can do any package manager operation on them from that screen.

You can press "g" to review the changes, modify the package additions and removals if necessary, and apply all pending operations.

A great visual way of package management over an ssh connection or plain old terminal.